Combining Like Terms Practice Problems

Mastering the Art of Combining Like Terms: Practice Problems and Solutions

Combining like terms is a fundamental algebraic skill crucial for simplifying expressions and solving equations. This comprehensive guide provides a thorough understanding of the concept, accompanied by a range of practice problems of varying difficulty. We'll start with the basics, gradually progressing to more complex scenarios, ensuring you develop a solid grasp of this essential mathematical technique. By the end, you'll be confident in identifying and combining like terms efficiently and accurately.

Understanding Like Terms

Before diving into practice problems, let's solidify our understanding of what constitutes "like terms." Like terms are terms that have the same variables raised to the same powers. The coefficients (the numbers in front of the variables) can be different, but the variables and their exponents must match exactly.

Here are some examples:

Like Terms: 3x and 5x (same variable, same exponent)
Like Terms: -2y² and 7y² (same variable, same exponent)
Like Terms: 4ab and -ab (same variables, same exponents)
Unlike Terms: 2x and 2y (different variables)
Unlike Terms: 3x² and 3x (different exponents)
Unlike Terms: 5ab and 5a²b (different exponents)
Like Terms: 6 and -2 (both are constants, meaning they have no variables)

Step-by-Step Guide to Combining Like Terms

Combining like terms involves adding or subtracting the coefficients of terms that are alike. Follow these steps:

Identify Like Terms: Carefully examine the expression and group together all terms with the same variables raised to the same powers.
Add or Subtract Coefficients: Add the coefficients of the like terms if they have the same sign, and subtract if they have opposite signs. Remember to pay attention to the signs (+ or -) in front of each term.
Write the Simplified Expression: Combine the results obtained in step 2 with the corresponding variable and exponent.

Practice Problems: Beginner Level

Let's start with some straightforward examples. Remember to follow the steps outlined above.

Problem 1: Simplify 2x + 5x

Solution: Both terms are like terms (same variable, same exponent). Add the coefficients: 2 + 5 = 7. The simplified expression is 7x.

Problem 2: Simplify 7y - 3y

Solution: Like terms. Subtract the coefficients: 7 - 3 = 4. The simplified expression is 4y.

Problem 3: Simplify 4a + 6a - 2a

Solution: Like terms. Add and subtract the coefficients: 4 + 6 - 2 = 8. The simplified expression is 8a.

Problem 4: Simplify 5b + 2 - 3b

Solution: Identify like terms: 5b and -3b. Combine them: 5b - 3b = 2b. The constant term 2 remains unchanged. The simplified expression is 2b + 2.

Problem 5: Simplify 8 - 5 + 3

Solution: All are constants (like terms). Combine: 8 - 5 + 3 = 6. The simplified expression is 6.

Practice Problems: Intermediate Level

These problems introduce slightly more complexity with multiple variables and different exponents.

Problem 6: Simplify 3x² + 5x - 2x² + x

Solution: Group like terms: (3x² - 2x²) + (5x + x) = x² + 6x

The simplified expression is x² + 6x.

Problem 7: Simplify 2ab + 4a - ab + 3a + b

Solution: Group like terms: (2ab - ab) + (4a + 3a) + b = ab + 7a + b.

The simplified expression is ab + 7a + b.

Problem 8: Simplify 5xy² - 2x²y + 3xy² + x²y

Solution: Group like terms: (5xy² + 3xy²) + (-2x²y + x²y) = 8xy² - x²y

The simplified expression is 8xy² - x²y.

Problem 9: Simplify 4x³ - 2x² + 7x³ + 5x² - x

Solution: Group like terms: (4x³ + 7x³) + (-2x² + 5x²) - x = 11x³ + 3x² - x

The simplified expression is 11x³ + 3x² - x.

Problem 10: Simplify -3p + 7q + 2p - 5q + 6

Solution: Group like terms: (-3p + 2p) + (7q - 5q) + 6 = -p + 2q + 6

The simplified expression is -p + 2q + 6.

Practice Problems: Advanced Level

These problems include parentheses and require applying the distributive property before combining like terms.

Problem 11: Simplify 2(x + 3) + 4x - 5

Solution: Distribute the 2: 2x + 6 + 4x - 5. Combine like terms: (2x + 4x) + (6 - 5) = 6x + 1

The simplified expression is 6x + 1.

Problem 12: Simplify 3(2y - 1) - 2(y + 4)

Solution: Distribute: 6y - 3 - 2y - 8. Combine like terms: (6y - 2y) + (-3 - 8) = 4y - 11

The simplified expression is 4y - 11.

Problem 13: Simplify 5(a² + 2a - 3) - 2(a² - a + 1)

Solution: Distribute: 5a² + 10a - 15 - 2a² + 2a - 2. Combine like terms: (5a² - 2a²) + (10a + 2a) + (-15 - 2) = 3a² + 12a - 17

The simplified expression is 3a² + 12a - 17.

Problem 14: Simplify x(x + 2) + 3(x² - 4)

Solution: Distribute: x² + 2x + 3x² - 12. Combine like terms: (x² + 3x²) + 2x - 12 = 4x² + 2x - 12

The simplified expression is 4x² + 2x - 12.

Problem 15: Simplify 2(3m - n) + 4(m + 2n) - 3(2m - n)

Solution: Distribute: 6m - 2n + 4m + 8n - 6m + 3n. Combine like terms: (6m + 4m - 6m) + (-2n + 8n + 3n) = 4m + 9n

The simplified expression is 4m + 9n.

Frequently Asked Questions (FAQ)

Q1: What happens if I have terms with different variables?

A1: You cannot combine terms with different variables. For example, 3x and 5y are unlike terms and cannot be simplified further.

Q2: What if a term doesn't have a coefficient?

A2: If a term doesn't have a visible coefficient, it's understood to have a coefficient of 1. For example, x is the same as 1x.

Q3: What about negative coefficients?

A3: Remember to include the negative sign when combining coefficients. For instance, 5x - 2x = 3x, and -7y + 3y = -4y.

Q4: Can I change the order of terms when simplifying?

A4: Yes, the commutative property of addition allows you to rearrange terms without changing the value of the expression. This can help organize like terms before combining them.

Q5: How do I check my answer?

A5: You can check your answer by substituting a value for the variable(s) into both the original expression and the simplified expression. If the results are the same, your simplification is correct. However, this method doesn't guarantee correctness for all possible values, only for the specific one you chose.

Conclusion

Combining like terms is a fundamental building block in algebra. By mastering this skill, you'll be better equipped to tackle more complex algebraic concepts. Remember the key steps: identify like terms, combine their coefficients, and write the simplified expression. Practice regularly using the problems provided (and create your own!) to build confidence and proficiency. With consistent effort, you'll effortlessly navigate the world of algebraic simplification. Don't be afraid to revisit these problems and challenge yourself to solve them again. The more you practice, the more intuitive combining like terms will become!